Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| automorphism | Aut(G) = #72256403 (order 2048) · Inn(G) = C2×B(4,𝔽2) · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×Q8 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
541 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 541 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13)(4,10)(5,18)(6,21)(7,12)(8,14)(9,17)(11,23)(15,28)(16,31)(19,25)(20,27)(22,24)(26,32)(29,30);(1,3)(2,8)(4,14)(5,19)(6,22)(7,15)(9,24)(10,28)(11,26)(12,25)(13,18)(16,23)(17,32)(20,30)(21,29)(27,31);(1,4,7,5)(2,9,12,27)(3,14,15,19)(6,20,23,17)(8,24,25,31)(10,11,18,21)(13,29,28,26)(16,32,22,30);(3,15)(8,25)(13,28)(14,19)(16,22)(24,31)(26,29)(30,32)Polycyclic presentation:
2^8|c^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,d^b=dh,e^b=eh,f^b=fgh,e^c=eg,f^c=fh,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152565, #153126, #153145, … |
| maximal normal subgroup of | #152565, #153126, #153145, … |
| maximal quotient of | #484177, #484182, #484263, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26557);